THE EFFECT OF SPEED OF ADVANCE UPON THE DAMPING OF HEAVE AND PITCH 3 
sponding positive square roots of these values. There are four 
such zeros in all and they are given by 
Ky, Ky = ky sec? O[1 + 2B cos @ + (1 + 4B cos 6)4] A 
3, K4 = $ Ko sec? O[1 — 2B cos 6 + (1 — 4B cos A)4] @) 
where ky = g/c?, B = p e/g, and x3, K4 only exist if cos 0 < 1/4 B. 
The integrals in equation (3) involving a factor of the form 
cos [u f (k)]/f(«) tend to zero as u —> 0, interpreting the inte- 
grals where necessary as principal value integrals. For the 
integrals in equation (3) of the form 
co 
j AcsialACMRG le 6 0 o ©) 
() 
where f («) has simple zeros, the contribution of each such zero, 
say ky, to the limiting value is 7 F (,)/|f’ («,)|. All the relevant 
zeros are included in the four values given in equation {4). 
After carrying out these operations, we obtain 
1 1 
g=m(——— 
Tale) 
m/2 ee) 
_2me!| ag sin (x x cos 8 + pt) 
7 (k ccos 8 — p)*? —gk 
0 0 
sin (k x cos 0 — pf) 
(x ccos@ + p)? —gk 
m/2 
2 Ky e—ki(d—z) 6 ; 9 P 
Mea (ROIS Icosgyiy ery +B ers y sin) 
0 
7/2 
2 Ky e— K2d—2) 9 here 
™ | O+4B cos 0 (kz x cos @ + pt) cos (kz y sin @) d 
0 
7/2 
sin pt 
Joos (x ysin 0) ke“K4-DJ dk 
Kee K(d—2) 
2m| a SACRO) Cos (k3 x cos 8 —pt)cos («3 y sin 8) dé 
61 
7/2 
2 Kg e7 Kad—z 9 
pall ciE=ataicosteynces Gece ze concarysin bat 
01 F 6 A 
where 
6,=0if4B<1 
HSCs CAM wassl , 5 o so @ 
In the last two integrals in equation (6) the integrand becomes 
infinite at the lower limit 6,, but the integrals remain finite in 
general; however, they become divergent in the limiting case 
when 4 B = 1 and 6, = 0. 
The wave pattern at a great distance from the source need not 
be discussed here; it is obtained by combining the last four 
terms in equation (6) with the suitable contribution from the double 
integral in equation (6). Broadly speaking, the pattern at a great 
distance in advance is associated with the «4 value while at the 
tear it comes from the k,, Ky and «3 terms. Finally, it can be 
verified that for c = 0, the expressions reduce to the known form 
for a stationary pulsating source emitting circular waves at a 
great distance. 
Consider now a long thin plank, of length L and beam B 
and with short pointed ends, floating vertically in water and 
immersed to a draught d. The plank moves forward with 
velocity c and makes small pitching oscillations with angular 
velocity Q sin p t. We assume that the velocity potential due to 
619 
the pitching motion can be derived from a source distribution 
over the flat base of the plank; further, we assume the source 
strength per unit area at a distance x from the mid-point is 
1/4 z times the normal velocity at that point, and for a sufficiently 
thin plank we take this as equivalent to a line distribution of 
amount (B/4 7) x Q sin pt. These are rather drastic simplifying 
assumptions, especially for pitching; but perhaps they are not 
too far amiss under the specified conditions for illustrating the 
particular point under consideration. To reduce the computa- 
tion we extend the integration only to cover the rectangular part 
of the base, omitting the supposed short pointed ends. The 
velocity potential due to the forward motion could be obtained 
in the usual way by a source distribution over the curved sides at 
the two ends of the plank; as this does not enter into the present 
calculation we omit this part of the velocity potential. 
Returning to equation (6) we obtain the required result by sub- 
stituting x —h for x, multiplying by h B/4 7 and integrating 
between the limits +/ for 4, where L = 2/. All the integrals 
can be evaluated explicitly, but to avoid lengthy expressions we 
write F (x, y, z) for the contribution of the first term in equation 
(6). We obtain thus 
¢ = (BQ/4 m7) F(x, y, z) sinpt 
7/2 
+ £BQ (E27) | do | (ic sec 8) Js ( 10s 8) 
0 0 
eS 
(« ccos 6 — p)? —gk 
uli cos (k x cos 0 — p t) 
(x ccos @ + p)* —gk 
| cos (k y sin 6) e~K@-) dk 
1/2 
13 \4 (x, sec 6)* : 
+ Ba —;) | (+4 Bos pel? (x, Lcos ) sin (kK, x cos 0 
0 
+ pt) cos («; y sin 8) e~@-) dO 
+ similar terms in kK, K3, K4 RUM ie Gee eee BS) 
where J denotes the ordinary Bessel Function. 
The pressure on the base is given by 
p=p(ot—c22) | eg 3G) 
ot ox 
and the moment M of the pressure about the axis O y is given by 
M= || pxdxdy eae eN ee GLO) 
taken over the base. Or, to the present approximation, 
I 
op d¢ 
M= pa (Ge =e. )xdx (11) 
with y = 0 and z = — d in equation (8). 
On examination of the various terms in equation (8) it is easily 
seen that the only contribution to the terms in sin pf in equa- 
tion (11) comes from the last four terms in equation (8). From 
the first of these terms, for instance, the contribution to this 
part of M is found to be 
7/2 
kK, ccos 0 — p 
+4 feos 6) 2? (,1cos O)e—214 sec 6 dO 
pe POsnps | 
0 (12) 
For computation we change from the Bessel Function J to the 
Spherical Bessel Function given by S (x) = (7/2 x)? J (&), 
